A rational expectations approach to macroeconometrics: Testing policy ineffectiveness and efficient-markets models 0226531864, 9780226531861, 9780226531922

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A Rational Expectations Approach to Macroeconometrics

A National Bureau of Economic Research Monograph

A Rational Expectations Approach to ｳ｣ｩｲｴ･ｮ ｯ ｣･ｯｲ ｡ｾ

Testing Policy Ineffectiveness and Efficient-Markets Models Frederic S. Mishkin

The University of Chicago Press

Chicago and London

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London

© 1983 by the National Bureau of Economic Research All rights reserved. Published 1983 Paperback edition 1983 Printed in the United States of America 90 89 87 86 85 543

Library of Congress Cataloging in Publication Data

Mishkin, Frederic S. A rational expectations approach to macroeconometrics. (A National Bureau of Economic Research monograph) Bibliography: p. Includes index. 1. Rational expectations (Economic theory) 2. Macroeconomics. 3. Econometrics. I. Title. II. Series. HB172.5.M57 1983 339'.0724 82-20049 ISBN 0-226-53186-4 (cloth) 0-226-53187-2 (paper)

NBER Board of Directors Officers Walter W. Heller, chairman Franklin A. Lindsay, vice-chairman Eli Shapiro, president David G. Hartman, executive director

Charles A. Walworth, treasurer Sam Parker, director of finance and

administration

Directors at Large Moses Abramovitz George T. Conklin, Jr. Morton Ehrlich Edward L. Ginzton David L. Grove Walter W. Heller Franklin A. Lindsay

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Bert Seidman Eli Shapiro Stephen Stamas Lazare Teper Donald S. Wasserman Marina v.N. Whitman

Directors by University Appointment Charles H. Berry, Princeton Otto Eckstein, Harvard Walter D. Fisher, Northwestern J. C. LaForce, California, Los Angeles Paul McCracken, Michigan Daniel McFadden, Massachusetts Institute

of Technology Almarin Phillips, Pennsylvania

James L. Pierce, California, Berkeley Nathan Rosenberg, Stanford J ames Simler, Minnesota James Tobin, Yale Willianl S. Vickrey, Columbia Dudley Wallace, Duke Burton A. Weisbrod, Wisconsin Arnold Zellner, Chicago

Directors by Appointment of Other Organizations Carl F. Christ, American Economic

Joel Popkin, American Statistical

Association

Association Robert C. Holland, Committee for

G. Edward Schuh, American Agricultural

Economics Association

EconOlnic Development Stephan F. Kaliski, Canadian Economics

Albert Sommers, The Conference

Board

Association Albert G. Matamoros, National Associa-

James C. Van Horne, American Finance

Association

tion of Business Economists Douglass C. North, Economic History

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of Certified Public Accountants

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Federation of Labor and Congress of Industrial Organizations

Directors Emeriti Arthur Burns Emilio G. Collado Solomon Fabricant Frank Fetter

Thomas D. Flynn Gottfried Haberler Albert J. Hettinger, Jr. George B. Roberts

Murray Shields Boris Shishkin Willard L. Thorp Theodore O. Yntema

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(Resolution adopted October 25, 1926, as revised through September 30, 1974)

To My Father

Contents

Acknowledgments

1. Part I 2.

3.

Part 2

Introduction

xi

1

Econometric Theory and Methodology The Econometric Methodology Appendix 2.1 Identification and Testing Appendix 2.2 An Annotated Computer Program An Integrated View of Tests of Rationality, Market Efficiency, and the Short-Run Neutrality of Aggregate Demand Policy

9 27 32

44

Empirical Studies

59

4.

Are Market Forecasts Rational?

5.

Monetary Policy and Interest Rates: An Efficient Markets-Rational Expectations Approach Appendix 5.1 Estimates of the Forecasting Equations Appendix 5.2 Additional Experiments Using the Two-Step Procedure

103

Does Anticipated Aggregate Demand Policy Matter?

110

6. ix

76 97

x

7.

Contents

Appendix 6.1 Output and Unemployment Models with Barro and Rush Specification Appendix 6.2 Results with Nominal GNP Growth and Inflation as the Aggregate Demand Variable Appendix 6.3 Results Not Using Polynominal Distributed Lags Appendix 6.4 Jointly Estimated Forecasting Equations

150

Concluding Remarks

156

References

159

Index

169

129 133 143

Acknowledgments

This book developed from a line of research that I have pursued for several years. In the process I built up an intellectual debt to many individuals who provided me with comments on my work and, by so doing, improved it substantially. My former colleague, Andrew Abel, is owed the greatest debt. Part of this book-Chapter 3 and Appendix 2.1 to Chapter 2-is based on joint research we worked on together at the University of Chicago. Andy not only stimulated my thinking in this line of research but also showed me how much fun joint work can be. I also thank the following other individuals who gave me valuable comments: Ben Bernanke, John Bilson, Olivier Blanchard, Edwin Burmeister, Dennis Carlton, Eugene Fama, Robert Flood, Jacob Frenkel, David Galenson, Peter Garber, Clive Granger, Nathaniel Gregory, Lars Hansen, Fumio Hayashi, Dennis Hoffman, John Huizinga, Stephen LeRoy, Robert Lucas, Thomas Mayer, Bennet McCallum, Merton Miller, Ronald Michener, Michael Mussa, A. R. Nobay, Charles Plosser, Thomas Sargent, Don Schlagenhauf, Williarrl Schwert, Steven Sheffrin, Robert Shiller, Kenneth Singleton, Gary Skoog, and Mark Watson. I benefited from comments at seminars where I presented preliminary versions of this work-at the American Economic Association winter meetings; Cornell University; the University of California at Berkeley, at Davis, and at San Diego; the University of Chicago; the University of Pennsylvania; the University of Michigan; the Universite de Montreal; the University of Virginia; the Massachusetts Institute of Technology; the National Bureau of Economic Research; and New York University. The students in my Economics 431 course during the Winter of 1982 at the University of Chicago performed the role of human guinea pigs by subjecting themselves to my teaching of this book. Their reactions and comments are greatly appreciated. xi

xii

Acknowledgments

I thank June Nason and Alyce Monroe for their typing services, and Alan Brazil and Douglas McTaggart for excellent research assistance. This book makes use of material from my articles published in American Economic Review, Journal of Finance, Journal of Monetary Economics, and Journal of Political Economy. I am grateful to each of these journals for permission to use material from these articles. Research support from the National Science Foundation (NSF grants SES-7912655 and SES-8112004) is gratefully acknowledged. This work is part of the National Bureau of Economic Research's Program in Economic Fluctuations. The usual disclaimer applies.

1

Introduction

The recognition in recent years that expectations are extremely important to economic decision making has led to a major revolution in macroeconomic analysis. The rational expectations hypothesis developed initially by Muth (1961) has played a critical role in this revolution. Simply put, it states that expectations reflected in market behavior will be optimal forecasts using all available information. (A more precise definition will be developed in Chap. 2.) When this hypothesis is employed to describe expectations formation, serious doubts arise about the use of existing large-scale macroeconometric models for policy analysis (Lucas 1976). With additional assumptions about labor market behavior, the effectiveness of any deterministic policy rule to promote macroeconomic stabilization is even called into question (Sargent and Wallace 1975). The rational expectations hypothesis has significant implications as well for the way macroeconometric models should be estimated. It leads to serious doubts about the traditional criteria for identifying and estimating these models-the exclusion of variables from some behavioral equations and not from others (Sims 1980). "(et we are not left helpless, because it does provide restrictions of a different sort which do allow identification and estimation, restrictions applying across equations (Sargent 1981). This book pursues a rational expectations approach to macroeconometrics, yet it is not a comprehensive treatment of the subject. Instead it explores a particular class of models, widely discussed in the macroeconomics literature, which emphasize the effects from unanticipated, rather than anticipated, movements in variables. In these models, crossequation restrictions implied by rational expectations are of central importance. These restrictions can be a powerful tool of analysis, making it 1

2

Introduction

easier to isolate the phenomenon we want to study. The following chapters will discuss and develop theoretically a unified econometric treatment of these models, and the resulting empirical analysis, based on Mishkin (1981a, 1981b, 1982a, 1982b, 1982c), will provide evidence on some of the more important macro issues debated today. How does the econometric methodology in this book fit into the overall subject of rational expectations macroeconometrics? Other econometric methodology recently developed is designed to analyze models where more structure has been imposed than in the class of models discussed here. For example, see Chow (1980), Hansen and Sargent (1980), Sargent (1978), Taylor (1979), and Wallis (1980), all of which can be found in the Lucas and Sargent (1981) collection. Hansen and Sargent (1980) provide a prominent example of this alternative approach to estimating rational expectations models. Their techniques can be used to look at the "deep structure" of economic relationships by estimating parameters describing tastes and technology. This allows the researcher to analyze problems that are out of reach of the techniques described in this book. However, there are advantages to the econometric approach here. Estimation is simple to execute with the techniques of this book and readily available computer packages, as the following chapter makes clear; this is less true of techniques such as Hansen and Sargent's. Fewer identifying assumptions are required to implement the econometric models analyzed here because the models are less structural. Because economists disagree strenuously about what is the appropriate structure of the economy (see the discussion in Sims 1980), empirical results obtained with fewer identifying assumptions are worth studying. The main conclusion to be drawn from these remarks is not that one set of econometric methodology is preferable to another; rather, all these techniques are needed for us to obtain a better understanding of how the economy works. Because the investment in reading a book is far greater than that in reading a short journal article, the reader deserves to be convinced that he or she will be involved in a productive activity. The first step in this process is to show that the methodology in this book is well worth studying because it is applicable to a wide range of research problems. With the increasing prominence of rational expectations in the last few years, there has been a veritable explosion in the number of empirical studies that either use or can use the methodology outlined in this book. These include studies of consumption behavior (Bilson 1980; Flavin 1981), the question whether government bonds are net wealth (Plosser 1982), the behavior of foreign exchange markets (Dornbusch 1980; Frenkel 1981; Hartley 1983; Hoffman and Schlagenhauf 1981b), the demand for money (Carr and Darby 1981), money and interest rates (Makin 1981; Mishkin 1981a, 1982c; Shiller 1980), money and stock

3

Introduction

prices (Rozeff 1974), the rationality of inflation and interest rate forecasts (Mishkin 1981b), asset returns and inflation (Bodie 1976; Fama and Schwert 1977, 1979; Jaffee and Mandelkar 1976; Nelson 1976; Schwert 1981), relative price variability and inflation (Fischer 1981), the effects of nominal contracting on stock returns (French, Ruback, and Schwert 1981), stock exchange seats as capital assets (Schwert 1977a) , regulatory effects of the securities exchange commission (Schwert 1977b), costs of financial intermediation and the Great Depression (Bernanke 1982), and the policy ineffectiveness proposition (Barro 1977, 1978, 1979; Barro and Hercowitz 1980; Barro and Rush 1980; Bjorklund and Holmlund 1981; Germany and Srivastava 1979; Gordon 1979; Grossman 1979; Hoffman and Schlagenhauf 1981a; Leiderman 1979, 1980; Makin 1982; Mishkin 1982a, 1982b; Sargent 1976a; Sheffrin 1979; Small 1979; Wogin 1980). This broad list of empirical topics above should persuade the reader that future research can profit from use of the methodology developed in the following chapters. However, the reader may still want to ask, What specifically is to be gained by reading this book? It is designed to be read on several levels and a brief "road map" of the analysis should help answer this question and save the reader both time and effort. The main body of the book is divided into two parts. Part 1 on econometric theory and methodology is the more technical. It 'Nill be of particular interest to the reader already convinced that the class of models analyzed in this book is worth studying, and who may want to pursue empirical work along these lines. The opening chapter of Part 1 describes the models to be analyzed in this book and discusses the details of their estimation. It covers such theoretical statistical issues as the consistency and efficiency of different estimation procedures. However, it is primarily a practical "how to" chapter that should help the interested researcher to apply the book's econometric techniques. To aid in the applications, an appendix to this chapter contains an annotated computer program that has been used to generate empirical results in Part 2 of the book. This program makes use of a standard econometric package available on many universities' computers and the techniques outlined here should be easily accessible to most researchers. Another purpose of this chapter is to provide sufficient information on the techniques used in the e:mpirical work in the following chapters, so that this work is opened to scientific scrutiny. This chapter, however, also is intended to be a fairly general treatment because the techniques discussed are applicable to many empirical issues not analyzed in this book. The next chapter treats theoretically the relationship among tests of rationality, market efficiency, and the short-run neutrality of policy. It provides information about how the tests conducted in this book relate to other tests in the literature, and it discusses the circumstances in which

4

Introduction

the resulting test statistics are generally valid. This chapter is useful in understanding both the empirical tests in this book and some of the previous literature on rational expectations models. Part 2 is much less technical and contains empirical studies on topics of potentially widespread interest. Chapters in Part 2 will be sufficiently self-contained so as not to require knowledge of Part 1 to be understood. Readers who are interested in the macroeconomic issues studied here but who are not technically inclined, or who are not yet convinced of the value of the econometric techniques, should read this section first. They may then find it worth investing in a careful reading of Part 1. The opening chapter of Part 2 provides information on the question of whether market forecasts are rational. Recent studies have found that forecasts of interest rates and inflation derived from well-known surveys are not consistent with the rational expectations hypothesis. For reasons discussed in this chapter, the survey measures may not reflect market behavior. We then want to know whether markets display rationality of expectations in contrast to the survey measures. If they do, doubt is cast on the survey measures as a description of market behavior. Tests similar to those conducted on survey data are used to provide evidence on this issue for the bond market. They do cast doubt on the usefulness of these survey measures and indicate for the most part that bond market forecasts are rational. Chapter 5 explores the relationship of monetary policy with both longand short-term interest rates. The impact of a money stock increase on nominal interest rates has been hotly debated in the monetary economics literature. The view most commonly held-also a feature of most structural macroeconometric models-has an increase in the money stock leading, at least in the short run, to a decline in interest rates. Monetarists dispute this view because they believe it ignores the dynamic effects of a money stock increase. The use of the rational expectations (or equivalently, the efficient markets) hypothesis in this analysis imposes a theoretical structure on the relationship that permits easier interpretation of the empirical results as well as more powerful statistical tests. In the interest of ascertaining the robustness of the results, many different empirical tests are conducted, and they do not support the proposition that increases in the money supply are correlated with declines in interest rates. Recent equilibrium business cycle models, which incorporate features of the natural-rate model with the assumption that expectations are rational, lead to an important neutrality result: Anticipated changes in aggregate demand will not evoke an output or employment response. One deterministic policy rule is, therefore, as good as any other from the point of view of stabilizing the economy. Chapter 6 conducts an empirical investigation of this policy ineffectiveness proposition. Not only do the

5

Introduction

results strongly reject the proposition, but, in addition, unanticipated movements in aggregate demand policy are not found to have a larger impact on output and unemployment than anticipated movements. The final chapter furnishes some concluding remarks and a general perspective on the empirical results and the econometric methodology.

PART

1

Econometric Theory and Methodology

2

The Econometric Methodology

2.1

The Models

The rational expectations hypothesis asserts that the market's subjective probability distribution of any variable is identical to the objective probability distribution of that variable, conditional on all available past information. Following the literature, we will restrict our attention to linear models and focus only on the first filoments of distributions; this yields models which are analytically and errlpirically more tractable. The rational expectations implication central to this book's analysis is the following: the expectation assessed by the market equals the true conditional expectation using all available past information. For a variable X, this can be written as (1)

where

Em ( E(

0 and Yt is represented as a time trend or a linear combination of such variables as the minimum wage and a measure of military conscription. Other empirical applications of this model include Barro 1979; Barro and Hercowitz 1980; Barro and Rush 1980; Bj6rkland and Holmlund 1981; Germany and Srivastava 1979; Gordon 1979; Grossman 1979; Hoffman and Schlagenhauf 1981a; Leiderman 1979, 1980; Makin 1982; Sheffrin 1979; Small 1979; and Wogin 1980. Following Modigliani (1977), this model will be referred to as the Macro Rational Expectations (MRE) model. The methodology discussed here is also worth studying for its useful applications in many recent empirical studies which analyze the differential effects of anticipated versus unanticipated movements in explanatory variables. These studies make use of the general model (8)

The Econometric Methodology

13

for different definitions of Yt, Yt, and Xt. They include Bernanke 1982; Bilson 1980; Bodie 1976; Carr and Darby 1981; Fama and Schwert 1977, 1979; Fischer 1981; Flavin 1981; Jaffee and Mandelkar 1976; Makin 1981; Nelson 1976; Schwert 1981; and Shiller 1980.

2.2

The Methodology

2.2.1

Estimation and Testing

The form of the efficient-markets equation is just a special case of the MRE equation where N = 0; hence the discussion here needs to focus only on the estimation and testing methodology for equations (7) and (8). To simplify the exposition, we will limit ourselves to the case where X t is a single variable. Modifications of the analysis for the case where X t is a vector of variables is straightforward. Rational expectations implies that the anticipations of X t will be formed optimally, using all available infornlation, and, as is usual in the literature, forecasting models are assumed to be linear. A forecasting equation that can be used to generate these anticipations is (9)

where Zt-l = a vector of variables used to forecast X t which are available at time t - 1 (this includes variables known at t - 1, t - 2, t - 3, ｾ

= Ut =

etc.), a vector of coefficients, an error term which is assumed to be uncorrelated with any information available at t - 1 (which includes Zt-l or Ut-l for all i 2:: 1, and hence U t is serially uncorrelated).

An optimal forecast for X t then simply involves taking expectations of equation (2) conditional on information available at t - 1. Hence (10) and, substituting into equation (7), we have N

(11)

Yt=Yt+ I ｾｬｘｴＭｩｚＩＫｅﾷ i=O

Two identification problems occur in the equation (11) model. Some assumption about the correlation of the error term, E, and the right-handside variables is necessary in order to identify the ｾ coefficients. The usual assumption-the one that is used in the tests here as well as in previous empirical work on this subject-holds that all the right-hand-side variables are exogenous and are uncorrelated with the error term. This

14

Econometric Theory and Methodology

assumption, that (11) is a true reduced form, implies that least-squares estimation methods will yield consistent estimates of the ｾＧｳＮ The other identification problem has been raised by Sargent (1976b). If Zt - 1 includes only lagged values of X t and there are no restrictions on the lag length N, the MRE model in (11) is observationally equivalent to "an unnatural rate model" where anticipated aggregate demand policy also matters. Hence, in this case, we cannot distinguish between the two competing hypotheses. To see Sargent's point, we can write the forecasting equation where only lagged X's are explanatory variables as, (12)

where 00

ｾＨｌＩ

= polynomial in the lag operator L = I i=1

ｾｩｌＮ

Taking expectations of (12) conditional on 4>t-l and substituting into (7) where N is not restricted, we have the MRE model 00

(13)

Yt = Yt + I ｾｩ｛ｘｴＭ

- ｾＨｌＩｘｴＭｩｬ｝

i=O

+ Et,

and this can be written as 00

(14)

Yt =Yt + I \fIiXt-i + Et, i=O

where \flo = ｾｯＬ

\fIi = ｾｩ

i-I

- .I ｾｪｩＭｬ

for ｩｾ

J=O

1.

If the forecasting equation in (12) is used to derive expectations in equation (8) where anticipated as well as unanticipated aggregate demand matters, we have 00

(15)

Yt=Yt+ I ｾｩ｛ｘｴＭＨｌＩｬ｝ i=O

00

+ I

i=O

8 i 'Y(L)Xt -

i- 1

+ Et

which also can be written as (14), where \flo = ｾｯ

\fIi = ｾｩ

i-I

+ .2 (8j J=O

ｾｪＩＧｙｩＭｬ

for ｩｾ

1.

Because both models can be written down as (14), the two models are observationally equivalent: that is, the data cannot discriminate between them because parameters are unidentified.

15

The Econometric Methodology

The problem of observational equivalence: has arisen in empirical work on whether anticipated aggregate demand policy matters, in particular, Grossman (1979). Grossman analyzes the MRE model where the aggregate demand variable is nominal GNP growth. His forecasting equation, however, includes only lags of nominal C..NP growth as explanatory variables. Because of the resulting observational equivalence problem, Grossman cannot and does not test whether the anticipated nominal GNP growth variables have significant additional explanatory power. Instead, he reports results supporting the MRE hypothesis which rely on flimsy grounds for identification, namely, the assumption that the lag length on nominal GNP growth cuts off at six quarters. It is possible to discriminate between the two competing models by means of identifying restrictions. These are derived by checking what conditions must be imposed to keep the MRE model and the model in which anticipated aggregate demand matters from being observationally equivalent. This exercise is carried out in i\.ppendix 2.1. The observational equivalence problem is overcome, parameters are identified, and tests of the MRE model are feasible, by either of two conditions particularly important in this book's empirical applications. They are: (1) N is known to be zero, as in the efficient markets model; or (2) Zt-1 includes lagged values of at least one other variable besides X which does not enter equation (11) separately from the ｾｩＨｘｴＭ -- Zt-1-i"l) terms. The method for estimating the MRE model involves joint, nonlinear estimation of the equations (9) and (11) system, which we rewrite as (16)

+ U t,

X t = Zt-1"1 N

Yt=Yt+ ｾ

i=O

ＨｘｴＭｩ ｚｴＭＱ ｩＧＩ ｾｩＫｅｴﾷ

System (16) embodies two sets of constraints. Rationality of expectations is imposed since the coefficient')' which appears in the equation for X t also appears in the equation for Yt. The neutrality property, that anticipated policy is not correlated with Yt - Yt, is also imposed because the 8 coefficients on Xr-i are constrained to be zero. Relaxing the neutrality and rationality constraints, the system (16) becomes (17)

X t = Zt-1')' + U t , Yt =Yt + ｾ

N

i=O N

+ I

i=O

(Xt - i - Zt-1-i')'*)f3i I

Zt-1-i"l*8 i + Et •

A likelihood ratio test comparing both the constrained system (16) and the unconstrained system (17) provides a joint test of both the rationality constraints')' = ')'* and the neutrality constraints 8i = 0, conditional on

16

Econometric Theory and Methodology

the maintained hypothesis of the model of equilibrium output. Note that (17) can also be written as (18)

X t = Zt-l'Y*

+ U t,

N

Yt =Yt + I ｾｩＨｘｴＭ i=O

N

- Zt-l-i'Y) + I '8 iX t - i + Et , i=O

where ｾ = ｾｩ - '8 i • This is the form used by Barro (1977) in his tests of neutrality. As an alternative to relaxing both the neutrality and the rationality constraints, we can relax one set of constraints only. For example, maintaining the hypothesis of rationality but relaxing the assumption of neutrality, system (16) becomes (19)

X t = Zt-l'Y

+ U t,

N

Yt=Yt+ I ＨｘｴＭｩｚｬＧｙＩｾＫ i=O

N

I Zt-l-i'Y'8 i + Et· i=O

Under the maintained hypothesis of rational expectations, the null hypothesis of neutrality, that is, '8 i = 0, can be tested by comparing the estimated systems (16) and (19). Rather than maintain the hypothesis of rationality of expectations and then test for neutrality, one can maintain the hypothesis of neutrality and then test for rationality. The unconstrained system used to perform this test is: (20)

X t = Zt-l'Y

+ U t,

N

Yt=Yt+ I ＨｘｴＭｩｚｬＧｙＪＩｾＫｅﾷ i=O

A comparison of the estimated systems (16) and (20) provides a test of the null hypothesis of rationality, that is, 'Y = 'Y*, under the maintained hypothesis of neutrality. In the efficient-markets case where N = 0, neutrality is a reasonable maintained hypothesis since the absence of neutrality would indicate the presence of unexploited profit opportunities. It must be noted, however, that a rejection of the null hypothesis that'Y = 'Y* may result from a breakdown of rationality, neutrality, or the model of market equilibrium. Furthermore, as is demonstrated in Appendix 2.1, when N = 0 this test is equivalent to that generated by comparing the systems (16) and (17), which jointly tests '8 0 = 0 and 'Y = 'Y* . The X2 statistic for the joint hypothesis of rationality and neutrality can be partitioned into the contribution from each component hypothesis by relaxing the constraints sequentially. These constraints can be relaxed in two different orders. A priori economic reasoning may suggest an appropriate sequence for relaxing constraints. For example, in testing whether anticipated policy is correlated with output, it seems appropriate

The Econometric Methodology

17

first to relax Ci == 0 and test neutrality under the maintained hypothesis of rationality. Then, without maintaining neutrality, the constraint ｾ == Ｊｾ can be relaxed, and rationality can be tested. This is the procedure that is followed in tests of the MRE hypothesis in Chapter 6. Under the alternative sequence for relaxing constraints, we first relax the constraint ｾ == Ｊｾ and test for rationality under the maintained hypothesis of neutrality. The next step in relaxing constraints permits a test of neutrality without maintaining the hypothesis of rational expectations. Yet neutrality has meaning only if we have a theory of expectations such as rational expectations. Realize that the test of neutrality is conducted on the assumption that the expectations of X t in the second equation of the system (17) are formed with the same information set Zt-1 as the time-series model of X t in the first equation. Yet, if we are not willing to assume that expectations are rational, there seems to be no reason to assume that the same set of variables belongs in Z in both equations in (17). Therefore, it is not clear that this test yields useful information. One way to generate the likelihood ratio statistics for the above tests is to estimate both the constrained and unconstrained systems with fullinformation-maximum-likelihood (FIML). Estimation proceeds under the identifying assumption used in previous research on the MRE hypothesis, that the y equation is a true reduced form. 1 This assumption implies that the covariance of the error term.s in the two equations of the system is zero. The estimated variance-covariance matrix of the residuals is then

SSR x == the sum of squared residuals of the X equation, SSR y == the sum of squared residuals of the y equation,

n

== the number of observations.

The systems analyzed here are triangular, so FIML involves maximizing the concentrated log likelihood function, (22)

log L

=

constant - ｾ log (det I),

1. In the case where N = 0, even if the asumption is untrue, its imposition will not invalidate the test statistics (see Chap. 3). However, it is not clear that this desirable result-that the assumption does not matter to the tests of interest here--earries over to the case where N > O.

18

Econometric Theory and Methodology

where det i (23)

=

determinant of i. The resulting likelihood ratio statistic - 2 log

LC(iC) ] - - A-

[ ｌｕＨｾ Ｉ

= nlog (det

ｾ｣Ｏ､･ｴ

A

ｾｕＩ

A

is distributed asymptotically as X2 ( q), where q = the number of constraints (Appendix 2.1 discusses how to count

them), L C= maximized likelihood of the constrained system, I: u = maximized likelihood o( the unconstrained system, ｾ｣ = the resulting estimated ｾ for the constrained system, ｾｵ = the resulting estimated ｾ for the unconstrained system. Comparison of this statistic with the critical X2 ( q) then tests the null hypothesis. Test procedures used in this book proceed in a slightly different way from that described above, in that they make use of nonlinear leastsquares estimation. Estimation is conducted with nonlinear least squares primarily for algorithmic reasons. FIML computer packages are usually not capable of handling large numbers of parameters, and this is required in some of the models analyzed empirically in this book. In addition, FIML packages do not allow us easily to impose the necessary covariance restrictions in (21) or to make a desirable degree of freedom correction, described below, which results in more conservative likelihood ratio statistics. In contrast, the nonlinear least-squares procedure outlined here easily implements the covariance restriction and degrees-offreedom correction and makes use of a computer package (SAS Institute 1979) that can estimate systems with large numbers of parameters. The procedure is as follows. Given an initial estimate for the variancecovariance matrix of the residuals, i, estimate the system with nonlinear generalized least squares (GLS). (The initial i can be obtained from unconstrained ordinary least-squares estimates of the X and y equations or from preyiously estimated systems.) Given the particular diagonal form of the ｾ matrix, nonlinear GLS is equivalent to nonlinear weighted least squares (WLS) using the estimates ｦｲｯｭｾＺ that is, the observations for the X forecasting equation are weighted by YSSRx/SSR y . Appendix 2.2 contains an annotated computer program describing this procedure in great detail. A new i matrix can be estimated using the resulting ｲ･ｾ siduals and the system reestimated again with nonlinear WLS. This iterative procedure is continued until there is little change in the i matrix. Because the system is triangular, this procedure will converge to maximum-likelihood estimates, since theorems showing that iterative threestage-least-squares is equivalent to FIML then apply to this nonlinear case as well. High computation costs required that iterations were continued only until the estimated variance of all the weighted equations in

19

The Econometric Methodology

the system differed by less than 5 percent. Some experimentation indicated that further iterations would have altered the likelihood ratio statistics in the book by at most 1 or 2 percent. This would only lead to a negligible effect on the inference drawn from these statistics. 2 If the same procedure is followed for estimating the unconstrained system, then the likelihood ratio statistic in (23) is easily calculated and can be used to test the null hypothesis. Although (23) yields valid asymptotic tests, it could be misleading in a small sample like that used here. The problem is that, in the maximum-likelihood calculation of the u matrix of the unconstrained system, no correction is made for substantial relative differences in the degrees of freedom in estimates of each unconstrained equation. Thus the finite sample distribution of the test statistic might differ substantially from the asymptotic distribution. For example, in Chapter 6, model2.1 (see tables 6.1 and 6.2), the unconstrained money growth equation is estimated with 79 degrees of freedom, while the unconstrained output equation is estimated with only 70 degrees of freedom. This is a difference of over 10 percent. The problem is even more severe in the case of model A16.1 (see tables 6.A.16 and 6.A.17) in Appendix 6.3 of Chapter 6: the degrees of freedom for the unconstrained money growth and output equations are now 79 and 32, respectively, a difference of 50 percent. Another way of stating this problem is to say that the weighting matrix for GLS will have a biased estimate of the variance of one equation relative to another. The bias occurs because the estimated variances are the maximum-likelihood estimates (the sum of squared residuals divided by the number of observations in each equation) rather than the unbiased estimates (the sum of squared residuals divided by the degrees of freedom). The likelihood ratio statistics reported here are corrected for the small sample problem as follows: the constrained system is estimated with the iterative procedure, and the resulting i c nlatrix from the constrained system is then again used with nonlinear WLS to estimate the unconstrained system. This corrects the degrees-of-freedom problem because, in the systems where there are cross-equation constraints, the degrees of freedom do not differ across equations. Thus i c does not suffer from the degrees-of-freedom problem of u • The resulting likelihood ratio statistic, which is also distributed asymptotically as X2 ( q) under the null hypothesis, is (Goldfeld and Quandt 1972)

i

i

(24)

- 2 log ｛ｕＨｾｃＩ｝

L U(!C)

= 2n log (SSRc/SSR U ) ,

2. In the empirical analysis in this book, when Goldfeld-Quandt (1965) tests revealed the presence of heteroscedasticity within an equation, the time-trend procedure outlined by Glesjer (1969) was used to weight each observation to eliminate this heteroscedasticity.

20

Econometric Theory and Methodology

where the superscripts on the I indicate that the maximized likelihoods of both the constrained and unconstrained systems were estimated with the same weighting matrix IC and SSRC = the sum of squared residuals from the constrained weighted system, SSR U = the sum of squared residuals from the unconstrained weighted system. Although asymptotically the two test statistics are equivalent, in finite samples the likelihood ratio statistic in (24) is smaller than the alternative in (23) and is more conservative on rejecting the null hypothesis. 3 To see this, realize that LU(IU) ｾ LU(IC), which implies (25)

- 2 10

g

LC(IC)] - -